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I have two Burny guitars and one Gibson with a 1960's slim neck. I'm just back from a gig and will be out all day tomorrow. I'll check on Monday for you when I'm back in the studio.
I would say they are very much in the 60s neck category.
The simple answer is that the '60s reissue Gibson neck on my ES-339 is slimmer than either of my Burny necks, the Burny Les Paul being just a bit thicker and the Burny jazz even thicker. I didn't have a measuring device, just my hands. You say you like a thicker neck; when I bought my ES-339 it was available with either a thicker '50s reissue neck or a thinner '60s reissue neck. You might prefer a '50s neck.
Chunky handful is a relative term. I consider my Guild 12 string with dual truss rods and my 5 string MusicMan Stingray bass to have necks that are chunky handfuls. Burny and Gibson are both less than a handful in my hands - but my hands may be a different size to yours.
As for an opinion on the necks, both Burny necks are much nicer to play than the Gibson (even though I prefer a thinner neck). This is mainly because Burny fretboards have a smoother finish than teh Gibson and when sliding between notes (with the fingers, not a slide), the Gibson frets are more vertical and act like a barrier trapping my fingers whereas Burny frets have a slope which allows fingers to slide over them. I would advise at the very least getting your hands on a Gibson and a Burny, regardless of the thickness of the neck, just to see which one feels better to you.